Power calculation of wavelength tunable Yb<sup>3+</sup>:LSO laser

Zhiyun Huang; Gaoming Li; Yishen Qiu

doi:10.1364/OE.18.020979

1. Introduction

An increasing number of Yb³⁺-doped oxyorthosilicate crystals have been reported in recent years, such as Yb³⁺:LSO (Yb³⁺:Lu₂SiO₅) [1–6], Yb³⁺:YSO (Yb³⁺:Y₂SiO₅) [1–4], Yb³⁺:SSO (Yb³⁺:Sc₂SiO₅) [2,3], Yb³⁺:GSO (Yb³⁺:Gd₂SiO₅) [7], etc. Their spectroscopic properties have been studied carefully. In these crystals, since Yb³⁺ occupies two different RE³⁺ sites, their spectrum exhibits the inhomogeneously broadened property, and the bandwiths of the absorption and emission spectrum are wide. The latter character enables Yb³⁺-doped oxyorthosilicate crystals to be used as the wavelength tunable gain media and the ultra-short pulse lasers. For example, the emission bandwidth of Yb³⁺:LSO is over 100 nm, and the laser wavelength tuning range is from 1025 nm to 1100 nm [1]. For Yb³⁺:LSO and Yb³⁺:YSO, the pulse duration could be as short as 260 fs and 122 fs, respectively [4].

Generally, the measured experimental absorption and emission spectra could not be regarded as the result of the single Stark level transition because of the thermal equilibrium distribution and the energy level broadening. The spectra obtained by experiment is the sum of the possible Stark level transition. As an example, in the emission spectrum of Yb³⁺:LSO crystal, the band around 1033 nm is the result of the transitions u ₁ → l ₄ and u ₃ → l ₆. In Yb³⁺:YAG crystal, the 1030 nm band could be treated as the result of the transition u ₁ → l ₃, as done in the previous models [8–13]. Thus the model about Yb³⁺:LSO laser is different from that about Yb³⁺:YAG laser. One flaw of these models is they could not be used to predict the wavelength tunable laser performance [8–13]. Peterson et al have developed the model about wavelength tunable Yb³⁺:YAG laser [14]. However, Yb³⁺ only occupies one Y³⁺ site in YAG host and exhibits the homogeneously broadened property. As well known, the laser property of inhomogeneously broadened medium is different from that of homogeneously broadened medium.

Then in this paper, a wavelength tunable Yb³⁺:LSO laser model is developed. The model is compared with the experimental results, and it verifies the validity of the model to some extent. Then by the proposed model, the impact of Yb³⁺ concentration and the crystal length on the wavelength tunable Yb³⁺:LSO laser performance is predicted.

2. Model

In Yb³⁺:LSO crystal, Yb³⁺ occupies two differente Lu³⁺ sites equally, and it gives rise to the wide absorption and emission bands [2]. In Fig. 1 , the energy level diagram of Yb³⁺:LSO is given.

Fig. 1 Energy level diagram of Yb³⁺:LSO crystal, where solid line represents the laser transition, dot line represents the pump transition, dash line represents the thermal redistribution.

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As shown in the figure, after Yb³⁺ ion of the ground Stark level (l ₁) absorbed the pump power, it transits to the corresponding Stark level of ²F_5/2 manifold, and then redistributs thermally between the Stark levels of two sites. By this mechanism, it seems the ²F_5/2 manifold is composed of 6 Stark levels and the ²F_7/2 manifold is composed of 7 Stark levels, and Yb³⁺:LSO crystal acts as the homogeneously broadened gain medium approximately. Because Yb³⁺ ion distributes equally between two sites, the degeneracy of the first Stark level of ²F_7/2 manifold (E_l ₁) should be regarded as 2.

Then the total population for ²F_5/2 manifold of two sites is governed by [15]

\frac{d N_{u} (ρ, z)}{d t} = \frac{I_{p} (ρ, z)}{h ν_{p}} σ_{p} Δ N_{p} (ρ, z) - \frac{I (ν, ρ, z)}{h ν} \sum_{i, j} σ_{i j} (ν) Δ N_{i j} (ρ, z) - \frac{N_{u} (ρ, z)}{τ_{f}}

Δ N_{p} (ρ, z) = f_{l 1} N_{0} - (f_{u 2} + f_{l 1}) N_{u} (ρ, z)

Δ N_{i j} (ρ, z) = (f_{u i} + f_{l j}) N_{u} (ρ, z) - f_{l j} N_{0}

where ρ and z are the radial and longitudinal coordinate. N_u(ρ,z) is the population of the upper manifold, ΔN_p(ρ,z) is the population participating in the pump process, ΔN_ij(ρ,z) is the inversion population between the ith (i = 1, 2, …, 6) Stark level of ²F_5/2 manifold and the jth (j = 1, 2, …, 7) Stark level of ²F_7/2 manifold. One should note that the degeneracy of the first Stark level of ²F_7/2 manifold (E_l ₁) is 2. I_p(ρ,z) and I(ν,ρ,z) are the pump and laser intensities depending on z location. h is Plank constant, ν_p and ν are the frequencies of the pump and laser beams resprectively. N ₀ is Yb³⁺ concentration, and τ_f is the fluorescence lifetime with concentration of N ₀. σ_p is the absorption cross section, σ_ij(ν) is the emission cross section at frequency ν and it represents the transition between the ith Stark level of ²F_5/2 manifold and the jth Stark level of ²F_7/2 manifold. f_ui and f_li are Boltzmann occupation factors of the ith Stark level in ²F_5/2 manifold and the jth Stark level in ²F_7/2 manifold, and they are calculated by

f_{u i} = \frac{g_{u i} \exp (- E_{u i} / k T)}{\sum_{i = 1}^{6} g_{u i} \exp (- E_{u i} / k T)}

f_{l j} = \frac{g_{l j} \exp (- E_{l j} / k T)}{\sum_{j = 1}^{7} g_{l j} \exp (- E_{l j} / k T)}

where g_ui and g_lj are the degeneracies of E_ui and E_lj levels. Here we should note that g_l ₁ = 2 and others equal 1. From Fig. 1, we could see 978 nm pump wavelength is corresponding to the transition of the zero line of site I. Also stated in Ref [1], 978 nm pump wavelength is optimum. Then in our model, we choose 978 nm as the pump wavelength, as Eq. (2)a) implies. From Eqs. (1), (2a) and (2b), we have

Δ N_{p} (ρ, z) = \frac{\frac{f_{l 1}}{τ_{f}} + \frac{I (ν, ρ, z)}{h ν} α (ν)}{\frac{1}{τ_{f}} + \frac{I_{p} (ρ, z)}{h ν_{p}} σ_{p} (f_{u 2} + f_{l 1}) + \frac{I (ν, ρ, z)}{h ν} β (ν)} N_{0}

Δ N_{i j} (ρ, z) = \frac{\frac{I_{p} (ρ, z)}{h ν_{p}} σ_{p} (f_{u i} f_{l 1} - f_{u 2} f_{l j}) + \frac{I (ν, ρ, z)}{h ν} [f_{u i} \sum_{i, j} f_{l j} σ_{i j} (ν) - f_{l j} \sum_{i, j} f_{u i} σ_{i j} (ν)] - \frac{f_{l j}}{τ_{f}}}{\frac{1}{τ_{f}} + \frac{I_{p} (ρ, z)}{h ν_{p}} σ_{p} (f_{u 2} + f_{l 1}) + \frac{I (ν, ρ, z)}{h ν} β (ν)} N_{0}

Δ N_{i j} (ρ, z) = \frac{f_{u i} f_{l 1} - f_{u 2} f_{l j}}{f_{u 2} + f_{l 1}} N_{0} - \frac{f_{u i} + f_{l j}}{f_{u 2} + f_{l 1}} Δ N_{p} (ρ, z)

where

α (ν) = \sum_{i, j} (f_{u i} f_{l 1} - f_{u 2} f_{l j}) σ_{i j} (ν)

,

β (ν) = \sum_{i, j} (f_{u i} + f_{l j}) σ_{i j} (ν)

. Inside the crystal, the pump and laser powers obey

\frac{d P_{p}^{\pm} (z)}{P_{p}^{\pm} (z) d z} = \mp 2 π \int_{0}^{\infty} σ_{p} Δ N_{p} (ρ, z) φ_{p} (ρ, z) ρ d ρ

\frac{d P^{\pm} (ν, z)}{P^{\pm} (ν, z) d z} = \pm 2 π \int_{0}^{\infty} σ_{i j} (ν) Δ N_{i j} (ρ, z) φ (ν, ρ, z) ρ d ρ - δ

where P_p ^± (z) and P ^± (ν,z) are the pump and laser powers along the positive and negative directions. P_p(z) and P(ν,z) represent the sum of the pump and laser powers of the two directions. δ is the internal loss per unit crystal length. ϕ_p(ρ,z) and ϕ(ν,ρ,z) are the intensity distributions of the pump and laser beams. For Gaussian beams, the intensity distributions of the pump and laser beams could be written as

φ_{p} (ρ, z) = \frac{2}{π w_{p}^{2} (z)} \exp [- \frac{2 ρ^{2}}{w_{p}^{2} (z)}]

φ (ν, ρ, z) = \frac{2}{π w^{2} (ν, z)} \exp [- \frac{2 ρ^{2}}{w^{2} (ν, z)}]

where w_p(z) and w(ν,z) are the spot radii of the pump and laser beams at location z. For plane-concave cavity, the values of w_p(z) and w(ν,z) obey

w_{p} (z) = w_{p 0} \sqrt{1 + \frac{ν_{p}^{2} z^{2}}{π^{2} w_{p 0}^{4}}}

w (ν, z) = w_{0} (ν) \sqrt{1 + \frac{ν^{2} z^{2}}{π^{2} w_{0}^{4} (ν)}}

where w_p ₀ and w ₀(ν) are the spot radii of the pump and laser beam waists. For pump beam, the value of w_p ₀ is determined by the pump source and the input configuration. For laser beam, inside the resonator, the value of w ₀(ν) is determined by the resonant cavity. It could be calculated by

w_{0} (ν) = \sqrt[4]{{(\frac{c}{π ν})}^{2} \frac{l (R_{i n} - l) (R_{o u} - l) (R_{i n} + R_{o u} - l)}{{(R_{i n} + R_{o u} - 2 l)}^{2}}}

where c is speed light, R_in and R_ou are the curvature radius of the input and output mirror, and l is the cavity length. From Eqs. (6)a) and (6b), the boundary conditions are:

P_{p}^{-} (z) = P_{p}^{+} (0) \exp [2 Γ (L) - Γ (z)]

P^{-} (ν, z) = P^{+} (ν, 0) \exp [- G (ν, z) - 2 δ z]

G (ν, L) = - \frac{1}{2} \ln (1 - T) - δ L

where Γ(z) = ln[P_p ⁺(z)/P_p ⁺(0)], G(ν,z) = ln[P ⁺(ν,z)/P ⁺(ν,0)], T is the transmission of the output mirror, and L is the crystal lenght. Here, the back-reflected pumping is considered and the reflectivity of the output mirror at the pump wavelength is assumed to be 1, i. e., the pump power is reflected totally at the output mirror. Inserting Eqs. (4)a) and (4b) into Eqs. (6)a) and (6b) gives

\frac{d Γ (z)}{d z} = - σ_{p} N_{0} \int_{0}^{1} \frac{\frac{f_{l 1}}{τ_{f}} + \frac{P^{+} (ν, 0) [e^{2 G (ν, z)} + e^{- 2 δ z}] α (ν)}{π w^{2} (ν, z) h ν e^{G (ν, z)}} y^{a}}{\frac{1}{τ_{f}} + \frac{P_{p}^{+} (0) [e^{2 Γ (z)} + e^{2 Γ (L)}] σ_{p} (f_{u 2} + f_{l 1})}{π w_{p}^{2} (z) h ν_{p} e^{Γ (z)}} y + \frac{P^{+} (ν, 0) [e^{2 G (ν, z)} + e^{- 2 δ z}] β (ν)}{π w^{2} (ν, z) h ν e^{G (ν, z)}} y^{a}} d y

\frac{d G (ν, z)}{d z} = N_{0} \int_{0}^{1} \frac{\frac{P_{p}^{+} (0) [e^{2 Γ (z)} + e^{2 Γ (L)}] σ_{p} α (ν)}{π w_{p}^{2} (z) h ν_{p} e^{Γ (z)}} y - \sum_{i, j} \frac{f_{l j}}{τ_{f}} σ_{i j} (ν)}{\frac{1}{τ_{f}} + \frac{P_{p}^{+} (0) [e^{2 Γ (z)} + e^{2 Γ (L)}] σ_{p} (f_{u 2} + f_{l 1})}{π w_{p}^{2} (z) h ν_{p} e^{Γ (z)}} y + \frac{P^{+} (ν, 0) [e^{2 G (ν, z)} + e^{- 2 δ z}] β (ν)}{π w^{2} (ν, z) h ν e^{G (ν, z)}} y^{a}} d y^{a} - δ

where a = w_p ²(z)/w ²(ν,z) and the boundary conditions are used. For the so-called mode matching, the approximations w_p(z)≈w(ν,z), ϕ_p(ρ,z)≈ϕ(ν,ρ,z), and a≈1 are adopted. Then by Eqs. (5), (6a), and (6b), we obtain

(f_{u 2} + f_{l 1}) σ_{p} [G (ν, z) + δ z] = σ_{p} N_{0} α (z) + β (ν) Γ (z)

Γ (L) = \frac{- (f_{u 2} + f_{l 1}) σ_{p} \ln \sqrt{1 - T} - σ_{p} N_{0} α (ν) L}{β (ν)}

For the given incident pump power and the resonant cavity, P_p ⁺(0), G(ν,L), and Γ(L) are known. Then for a guess P ⁺(ν,0), G(ν,L) could be solved numerically by Eq. (11)a) or (11b). If the solved G(ν,L) is in agreement with that derived from the boundary condition, the output power is P_o(ν) = P ⁺(ν,0)exp[G(ν,L)]T. Otherwise, another P ⁺(ν,0) is assumed and the calculation process continuous.

3. Results and discussions

The emission spectrum of Yb³⁺:LSO crystal are fit by the superimposition of six Lorentzian transitions, since Yb³⁺:LSO acts as the homogeneously broadened gain medium, and the results are given in Table 1 . It seems the fit bands around 999 nm, 1033 nm, and 1083 nm are resulted from two transitions, and so the emission cross section of the band should be regarded as that of the sum of two transitions. It is difficult to determine each emission cross section of two transitons. Then for the sake of simplicity, the emission cross section of each transition is regarded as the half emission cross section of the band. This approximation may deviate from the truth. However, as shown in the following comparison between the experimental and the calculated results, the deviation is acceptable.

Table 1. Analytical results of Yb³⁺:LSO emission spectrum^a

View Table

Firstly, the model is applied to the experiment in order to verify the validity of the model. The experimental data are taken from other group [1]. The calculated and experimental results are plotted in Fig. 2 , and the values of the experimental parameters are [1]: P_p ⁺(0) = 14 W, λ_p = 978 nm, σ_p = 1.0 × 10⁻²⁰ cm², τ_f = 0.95 ms [2], N ₀ = 8 at.%, 1 at. % = 1.95 × 10²⁰ cm⁻³, L = 2 mm, w_p ₀ = 100 μm, T = 0.04, and δ = 0.01 cm⁻¹. Here the absorption cross section is calculated directly from the measured absorption coefficient, and X polarized emission cross section is adopted in the calculation. The experimental curve is taken from Fig. 6 of Ref [1]. As shown in this figure, for the experimental results, the wavelength tuning range is from 1026 nm to 1098 nm, and the power peaks locate at 1057 nm, 1068 nm, and 1083 nm; for the calculated results, the wavelength tuning range is from 1026 nm to 1095 nm, and the power peaks locate at 1044 nm, 1057 nm, and 1083nm. For the tuning range, it seems the experimental and theoretical results are almost the same. Moreover, the bandwith of more than 4 W output is almost the same for experimental and calculated results. However, for the output power, the difference between the experimental and theoretical results is a little large.

Fig. 2 Calculated and experimental output power, where P_p ⁺(0) = 14 W, N ₀ = 8 at.%, L = 2 mm, w_p(0) = 100 μm, T = 0.04, and δ = 0.01 cm⁻¹.

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The difference may come from some factors which are not considered in our model. One possible factor is the thermal effect. As well known, the heat generation is unavoidable during the laser operation and impacts the laser performance. As can be seen from the absorption and emission spectra [1,2], the preferential lasing axis is X axis. In our calculation, X polarized emission cross section is adopted. In the experiment, Yb³⁺:LSO crystal is placed with X axis in the plane of cavity [1]. Inside cavity, no linear polarizer is inserted, and the output power may not be linearly polarized. This may also account for the discrepancy. Finally, the more accurate analysis of the emission spectrum may help to improve the calculated result. As shown in the first paragraph of section 3, the emission cross section of the band around 999 nm, 1033 nm, and 1083 nm should be regarded as that of the sum of two transitions. But in our calculation, since it is difficult to distinguish these two transitions, the emission cross section of each transition is regarded as the half emission cross section of the band.

After the validity of the model is verified, the tunable laser performances of Yb³⁺:LSO could be predicted. In Fig. 3 , the laser performance is calculated at different Yb³⁺ concentration, where P_p ⁺(0) = 10 W, T = 0.01, L = 1 mm, δ = 0 cm⁻¹, and N ₀ = 5 at.%, 6 at.%, and 8 at.%. For these calculations, the fluorescence lifetimes are assumed to be 0.95 ms for three cases since the variation of the concentration is not large. As can be seen in the figure, of all the concentrations considered, the laser performance for 8 at.% sample is the best because its wavelength tuning range is the widest and its output power is the highest. However, for the given crystal length, we could not conclude from this figure that high Yb³⁺ concentration is good for the laser performance. If Yb³⁺ concentration is too high, the reabsorption and the possible concentration quenching are serious, and these two factors reduce the output power.

Fig. 3 Calculated output powers for different Yb³⁺ concentrations, where P_p ⁺(0) = 10 W, T = 0.01, L = 1 mm, δ = 0 cm⁻¹, and N ₀ = 5 at.%, 6 at.%, and 8 at.%.

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Figure 4 shows the laser performance varying with the crystal length, where P_p ⁺(0) = 10 W, T = 0.01, N ₀ = 8 at.%, δ = 0 cm⁻¹, and L = 0.5 mm, 1 mm, 2 mm, and 3 mm. Since Yb³⁺ laser is quasi-three level system, too long crystal length is not very good for the laser performance, as shown in Fig. 4. Of all the three lengths, for 3.0 mm, the wavelength tuning range is the narrowest because of the reabsorption of Yb³⁺ ion. For the short crystal length, as the curve of 0.5 mm shows, the wavelength tuning range is not wide either, because 0.5 mm crystal length only absorbs about 50% incident pump power for single pass and about 80% for reflected pass. Then in order to obtain the good laser performance, the right crystal length should be chosen.

Fig. 4 Plots of the calculated output powers for different crystal length, where P_p ⁺(0) = 10 W, T = 0.01, N ₀ = 8 at.%, δ = 0 cm⁻¹, and L = 0.5 mm, 1 mm, 2 mm, and 3 mm.

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The impact of the transmission of the output mirror on the laser performance is plotted in Fig. 5 , where P_p ⁺(0) = 10 W, N ₀ = 8 at.%, L = 1 mm, and T = 0.001, 0.005, 0.01, 0.02, and 0.03. Low transmission gives rise to the low cavity loss and the low threshold as well as the wide wavelength tuning range. At some wavelengths, for low transmission, the reflectivity of the output mirror is not large enough to extract the laser power from the resonant cavity, which lowers the output power. High transmission means the high cavity loss, which lowers the output power and narrows the wavelength tuning range. As shown in Fig. 5, the tuning range of T = 0.001 is the widest, and that of T = 0.03 is narrowest.

Fig. 5 Calculated output powers for different transmission of the output mirror, where P_p ⁺(0) = 10 W, N ₀ = 8 at.%, L = 1 mm, and T = 0.001, 0.005, 0.01, 0.02, and 0.03.

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4. Conclusion

Compared with Nd³⁺ crystal, the emission bandwith of Yb³⁺ crystal is wide. This property enables Yb³⁺ crystal to be used in the fields of wavelength tunable laser and short pulse laser. Though some models about Yb³⁺ laser have been developed, they are built to treat the homogeneously broadened lasers. None of them could be used to calculate the wavelength tunable Yb³⁺:LSO laser. In this paper, a numerical model is developed for wavelength tunable Yb³⁺:LSO laser, which takes into account the spectral information of the laser transition.

The calculated result agrees with the experimental result to some extent. The predictions about the impact of Yb³⁺ concentration, the crystal length, and the transmission of the output mirror on the wavelength tunable Yb³⁺: LSO laser performance are given. In Fig. 3, it seems 8 at % is the best concentration in the three concentrations considered. As shown in Fig. 4, of all the crystal length considered, it may be difficult to determine which crystal length is the best. However, we could still conclude the laser performances of 1 mm and 2 mm are better than those of 0.5 mm and 3mm. In Fig. 5, of all the transmission of the output mirror considered, it seems 0.001 is suitable though in some wavelength, its output power is not the highest. Therefore these calculation implies that to obtain the satisfactory laser performance, the optimization about Yb³⁺ concentration, crystal length, and the transmission of the output mirror needs to be performed. In fact, for Yb³⁺ lasers, such as Yb³⁺:YAG laser, it exists the optimum Yb³⁺ concentration, the optimum crystal length, and the optimum transmission of the output mirror.

The model could be applied not only to wavelength tunable Yb³⁺:LSO laser, but also to other wavelength tunable Yb³⁺-doped oxyorthosilicate lasers.

Acknowledgements

This work is supported by the Fund of Key Laboratory of Optoelectronic Materials Chemistry and Physics, Chinese Academy of Sciences (2008DP173016) under grant 2010KL0014.

References and links

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Center wavelength λ_c (nm)	FWHM Δλ (nm)	Amplitude A (10⁻²⁰ cm²)	Transition
978.9	5.4	3.3	u₂ → l₁
991.4	5.2	1.5	u₁ → l₂
999.1	42.7	41.0	u₂ → l₃ and u₄ → l₄
1032.7	18.0	9.4	u₁ → l₄ and u₃ → l₆
1041.2	9.3	2.7	u₁ → l₅
1055.6	15.2	10.0	u₁ → l₆
1083.0	21.8	4.8	u₁ → l₇ and u₂ → l₇

Power calculation of wavelength tunable Yb³⁺:LSO laser

Abstract

1. Introduction

2. Model

3. Results and discussions

4. Conclusion

Acknowledgements

References and links

Cited By

Figures (5)

Tables (1)

Equations (22)

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